A Beautiful Dream
Suppose, one day, you stray off your usual path and wander off into some out-of-the-way place. After a time, you realize you're lost. Looking around for someone who can show you the way back, you come across me. You're about to ask me for directions, but then you notice that in front of me is a fantastic machine the likes of which you have never seen before.
Naturally, you ask me, "What does that machine do?"
"It's a truth machine," I say.
This sounds outrageously impossible, but you're intrigued enough that, with some coaxing from me, you put aside your objective of getting directions and decide to ask the machine a question. So you think of an easy question and we feed it into the machine. Lo and behold, it comes back true.
"Ah, well that was too easy," you say.
"Well, ask as many questions as you want," I reply.
So, we proceed to feed some harder questions into the machine. Each one seems to come back with a truthful and accurate answer. Emboldened, you start to ask the machine some really wild and speculative questions. These come back with answers that are far too big to verify right away, but you're really intrigued now, and you want to make sure this isn't a scam or a prank.
"How does it work?" you ask me.
"It's very simple," I say.
I proceed to give you a thorough and convincing explanation of the basic scientific principles at work, and the engineering methods I've applied to them in order to construct the machine. (The details of this explanation are left as an exercise to the ambitious reader.) It becomes clear that the machine is not just a high-class Mechanical Turk, or a nice bit computational ventriloquism on my part. Moreover, there's no sorcery, no divine intervention, and nothing weird or unexplained. The workings of the machine are unambiguous consequences of straightforward and readily accessible scientific facts, and they are all perfectly clear to you.
"That's amazing!" you say.
You're overcome with a wave of excitement and decide to get back to the rest of the world so you can test out the machine's answers to your really difficult questions. I give you directions back to the beaten path, and you go on your way. Perhaps, even more interestingly, it is the Machine that gives you the directions.
I might try to tell the rest story but, as wild and fantastical daydreams go, the uses and consequences of the Truth Machine would far surpass even the Machine itself. Would all those beautiful utopian dreams of the Enlightenment finally come true? Or would we turn our knowledge destructively in on ourselves and extinguish the entire universe? Would truth become another commodity to be bought, sold and advertised on the market? Or would the accumulated knowledge of humanity finally be free to anyone curious enough to ask? I won't speculate, because speculation is not my purpose here.
Instead, consider a slight variant of the scenario above.
A Variation on a Theme
Suppose that everything happens as before; we meet, and I demonstrate the Truth Machine. After you've had your fill, we part ways, but you have so many more questions that you decide to come back the next day. You find me and the Machine in the same place, and can hardly contain your excitement.
"I have more questions," you say. "Can I ask the machine?"
"Well, I'd love to oblige," I reply, "but it's too hot today. The heat will interfere with the way the machine works, and it will give false output."
I go on to explain why this is, and it's immediately clear that this is not only consistent with the previous day's explanation, but a necessary consequence of the way the machine works.
You turn to leave, but you're very clever and perceptive, and it occurs to you that this kind of situation often gives rise to paradoxes or infinite regressions, or other such weirdness. Deciding to really put the machine to the test, you turn back to me and propose the following:
"Suppose I ask the machine a question today, take the answer I get, and come back on some day when the machine is working and ask 'How would the Truth Machine answer such-and-such question on a day when the temperature such-and-such many degrees?' If you really do have a truth machine here, and if the ambient temperature does affect its function, then even though the machine answers the question wrong today, it should be able to reproduce and explain its wrong answer later, on some day that it is working."
"That's true," I say. "Why don't you ask your question, write down the answer, and come back later when the machine is working again."
And so, you ask your question and write down the answer. A few days later, you come back, and together we ask the machine how it would respond to your earlier question on a hot day. Much to our surprise, nothing weird or paradoxical happens: the machine correctly reproduces its wrong answer, and gives an accounting for its failure. We're both so thoroughly amazed that you become totally convinced that the Truth Machine works as promised, and I suffer a minor injury from patting myself on the back for my accomplishment.
Some Casual Dream Analysis
There are several ways to read this variation of the story. The story's description of the machine replicating and explaining its own mistakes is something like what happens when one scientific theory subsumes another: a new scientific theory must always explain why its predecessor seemed to be true. This is one interesting feature.
Another interesting feature is brought out if we continue to vary the story by adding additional constraints on the correct function of the machine. For instance, suppose that the machine is sensitive not only to temperature but also, say, electrical fields so that a passing storm cloud might disrupt its function. We can go further and suppose that the machine is sensitive to temperature, electrical force, and perhaps noise. Maybe certain levels of background radiation have an effect. Perhaps very slight changes in gravitational force at the point on the Earth's surface where we are running the machine, due to elevation or the moon's orbit, may also disrupt its function. This doesn't seem unreasonable; after all, the Truth Machine must be extremely delicate. It would seem that we could go on adding necessary constants indefinitely (if you say that we couldn't, then it seems you know more about truth engineering than me) until running the truth machine and getting a valid response requires carefully planned and controlled conditions. Of course, the machine still gives us correct answers; it's just that we must be very careful that exactly the right external conditions are in place to ensure that all the necessary processes within the machine work as expected.
What do we get if we add enough constraints? At some point, the Truth Machine will start to look less like a stunning technological oracle and more like an ordinary experimental apparatus, in the spirit of Boyle's air pump or the Large Hadron Collider. Of course, our fictional construction here differs from typical experimental apparatuses in that it seems we can pose it any question we please and, given the right circumstances, get a correct reply. By contrast, we may be able to ply the LHC for answers about elementary particles or black holes, but it is not immediately apparent how it could give us direct answers about how animal cells specialize into organs, whether the Navier-Stokes equations are a faithful model of fluid dynamics, or how exactly a pair of sociopathic teenagers burning a cat to death in an Internet video could give rise to the NEDM phenomenon. This is an important distinction. Most experimental machines are constructed to answer one specific question, or a specific class of questions.
(On the other hand, if it's perfectly clear to you how the details of elementary particle theory relate to Internet memes, you should really be talking to the Nobel Committee.)
What I've sketched out here is something like the outline of traditional scientific knowledge. Truth comes from relating reproducible phenomena that we do understand to other phenomena that we know less about. In this case, we leveraged what we knew about the workings of the machine into new information that we didn't know. I've purposefully resisted the strong temptation to involve exotic, modern abstractions like algorithmic information theory (Would our machine have to contain all the information in the history of the Universe?), or classic tropes like a Turing-style non-termination scenario (What happens if we ask the machine to predict the future of its own execution?), but not because I think such considerations are irrelevant. Mathematical logic has produced some truly strange and remarkable artifacts in the last century, and as deserving of recognition as these are, it is not immediately clear how they relate to the physical world as we know it. If I brought such abstract machinery into play here, I feel that I would be admitting unfounded back-door assumptions about how the physical world in general, and Truth Machines in particular, must work.
However, I'm not done just yet. Supposing that we do obtain a context-sensitive Truth Machine as described above, is the only work left for the thinkers and discoverers of the world to work out a calculus ratiocinator and start dreaming Leibniz's dream?
Dream On, Leibniz
We meet, I demonstrate the Truth Machine. I explain that it doesn't work on hot days (or stormy days, or at high altitudes, etc. etc.), and you're satisfied that this is indeed a necessary consequence of the appropriate Laws of Nature. As before, you ask the Machine a question on a hot day, in order to really put the Machine to the test by checking the consistency of its answers. Realizing that it may be important what question to ask, you think for a while, and then come up with something.
"Let's ask these two questions," you say. "(1) Is A true? [where A is any meaningful proposition] (2) How will a truth machine answer the question 'Is A true?' under conditions suitable for a correct answer'"
"Well that's a weird question," I say, "but alright. Let's feed it into the machine."
We ask the Machine your question. The Truth Machine gives the following answer: "A is false [without loss of generality, since we can transform any true proposition into a false one by negation], and under conditions suitable for a correct answer, a truth machine will find that A is false." I suggest that you write down the answer and come back tomorrow when the machine will be working again, but you're not satisfied just yet. You go on to ask the machine another question:
"What are necessary conditions for a truth machine, as constructed here, to give correct answers, and why are these conditions necessary for a correct answer?"
The machine responds by describing exactly the physical conditions under which it is operating at the moment. It goes on to give a response that explains, at great length and with considerable detail, why these conditions are essential to producing a correct answer. You and I both seem to understand the response, and after looking over it carefully, neither of us is able to find any internal inconsistency to it. Moreover, the truth or falsity of A is not immediately verifiable without the Machine's help.
"Well," I say, "the machine isn't working right anyway. Of course it would say that. Come back tomorrow, and we'll see what it says."
The next day, you return and we repeat your questions to the machine. According to my earlier claims, conditions are favorable to truth machines that day, and the machine gives the following response: "A is true, and under conditions suitable for a correct answer, a truth machine will find that A is true."
"There you have it," I say, " I guess A is true after all."
But you've become a little more skeptical now. You decide to repeat your other question as well, and you ask the Machine a second time:
"What are necessary conditions for a truth machine, as constructed here, to give correct answers, and why are these conditions necessary for a correct answer?"
As before, the machine responds by describing exactly the physical conditions under which it is operating at the moment. It goes on to give a response that explains, at great length and with considerable detail, why these conditions are essential to producing a correct answer.
"And that's what I said before," I volunteer. "It's all settled."
But you're really troubled by all this now. A shadow's fallen over the enterprise, and although you're not prepared to dismiss me or the Machine out of hand, something seems unaccounted-for.
"How do we know that it wasn't working correctly yesterday?" you say. "Maybe you're wrong about the external conditions, and the machine is actually malfunctioning today."
"But who do we trust? You? Me? The Machine?"
"Brother, let us reason."
Whose claims should we trust? Presumably, we're all interested in nothing more than getting to the truth of the matter. We've all sworn "Calculemus!", true to the spirit of ol' Leibniz, and we're willing to trust the Machine so long as we can verify that it's working as expected. Still, there is a clear problem of which source of assertions to trust. The Machine throws things out of balance because, presumably, it is a new and very large source of information that has come into conflict with a body of information that, as far we know, is well-founded and correct.
This conflict is distantly reminiscent of Poincare's skepticism toward Frege's project of formalizing all knowledge and placing it on a clear and unambiguous axiomatic foundation[1]. Frege seemed to believe in the possibility of realizing a genuine calculus ratiocinator, whereas Poincare argued that the soundness of such a system could never be empirically proven, since it would require the verification of infinitely many theorems. One could always argue for a sort of "generative proof" of the sort one sees in formal logic: show that the axioms of the system do not conflict with one another, then show that it is impossible for any application of the proof rules to produce a contradiction, whence any theorem proved from the axioms cannot be a contradiction. This is fine for an abstraction but, Poincare points out, our abstraction is of interest only to the extent that it reflects some material reality, and no proof of internal soundness can guarantee the faithfulness of a model. Phrased another, we don't know if or when an empirical contradiction might jump out and surprise us.
How can we decipher the conflicting responses of our troublesome Machine? The problem arises because the Machine seems to have distinct and incommensurate frames of reference: it answers the same question one way on one day, and the other way on a different day. One way we might get to the bottom of all this would be to keep plying the Machine with questions, both on "on"-days and on "off-"days to see if we could catch it in a contradiction. If the machine gave a reply that was plainly false to us, the observers, or that contradicted an earlier response we would know that the external conditions under which the machine was operating could not be the right circumstance for truth machines to be truthful. However, there seems to be no way to tell how many questions we might have to ask the machine before we caught it in a contradiction; we might question the Machine until the end of time and never see anything but self-consistency.
As long as the machine continues to explain its own operation and to do so in a way that reveals no contradictions that aren't directly tied to changes in operating conditions, we have no reason to believe that its answers are anything other than the necessary consequences of essential Laws of Nature.
Worse still, there's no way we could ask the machine for help in the matter, since we've already cast doubt on its judgments.
Who do we trust? You? Me? The Machine? And how should we understand the Machine's explanations of its own contradictions?
Though I won't pretend that these questions are impossible or intractable, I also don't see an obvious answer. There is, however, one last variant that may say something important, though it may not resolve this particular dilemma.
Ground Zero
Recall that in the paragraphs above, we took a completely idealized and perfect Truth Machine and replaced it with one that was affected by its physical operating conditions. We observed that we could elaborate a succession of factors that might all contribute to the machine's correct operation, and that so doing lead to something reminiscent of an ordinary experimental apparatus. In schematic, we have a succession like so:
(1) Given input A, the machine produces output B and B is true.
(2) Given input A, the machine produces B and B is true if the machine doesn't get too hot.
(3) Given input A, the machine produces B and B is true if the machine doesn't get too hot and if the machine is insulated from strong electric forces.
...
And so on, until we've taken into account all of the factors that might affect the Machine's operation. This is really a sequence of refinements to our idea of of the variables upon which the machine's output depends. That is, we could look at our succession as:
(1) The machine's output depends on its input.
(2) The machine's output depends on its input and the ambient temperature.
(3) The machine's output depends on its input and the ambient temperature and net electric forces acting on its mechanisms.
(4) The machine's output depends on its input and the ambient temperature and net electric forces acting on its mechanisms and ...
And so on. Looked at this way, it seems that there's one variable that we've forgotten, in our rush to account for necessary operational constants. Consider what we get if we take a step backward from (1):
(0) There is output.
Even the ideal machine we started with had one factor that affected its operation, namely, the question that its user asked. We began to doubt the Machine once its operating conditions allowed it to give different answers and also to account for them. However, we would in general expect our machine to give different answers to different questions. (Unless, of course, it just "holds up a finger."[2]) If we hold that (1) humans are physical beings, and (2) their actions comprise physical phenomenon, and (3) asking a question to the Truth Machine is a physical phenomenon, then since (4) the Machine's output depends upon the question it is asked, it must be that a question is an external physical condition affecting the operation of the Machine. This leads us to one last question:
Is is it function or malfunction for the Truth Machine to give different answers to different questions?
[1] Poincare, Henri, Francis Maitland (Trans.). Science and Method, Barnes & Noble Publishing, 2004 (original 1908).
[2] See case 19 of The Blue Cliff Record
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